Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space

Pages: 555 - 582, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.555

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Jihong Zhao - College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China (email)
Ting Zhang - Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email)
Qiao Liu - Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China (email)

Abstract: In this paper, we are concerned with a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. The local well-posedness and global well-posedness with small initial data to the 3-D Cauchy problem of this system are established in the critical Besov space $\dot{B}^{-1+\frac{3}{p}}_{p,1}(\mathbb{R}^{3})\times(\dot{B}^{-2+\frac{3}{q}}_{q,1}(\mathbb{R}^{3}))^{2}$ with suitable choices of $p, q$. Especially, we prove that there exist two positive constants $c_{0}, C_{0}$ depending on the coefficients of system except $\mu$ such that if \begin{equation*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+(\mu+1)\|(v_{0},w_{0})\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{\frac{C_{0}}{\mu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+1)\Big\}\leq c_{0}\mu, \end{equation*} then the above local solution can be extended to the global one. This result implies the global well-posedness of this system with large initial vertical velocity component.

Keywords:  Navier-Stokes equations, Poisson-Nernst-Planck equations, electro-hydrodynamics, well-posedness, Besov space.
Mathematics Subject Classification:  Primary: 35K15, 35K55; Secondary: 35Q35, 76A05.

Received: November 2013;      Revised: June 2014;      Available Online: August 2014.